Elements of combinatorics. Division by a column of multivalued natural numbers
LESSON SUMMARY
MATHEMATICS
3rd grade
Plokhotnyuk Viktoria Nikolaevna,
teacher primary school
MBOU "Secondary School No. 6", Usinsk
Republic of Komi
TOPIC: Repetition of division (reception of calculating the quotient)
TASKS:
to continue work on the division technique based on operating with specific objects;
fix the name of numbers when dividing, multiplying;
develop the skill of oral counting;
keep working on communication skills
DURING THE CLASSES:
The lesson starts.
He will go to the guys for future use.
I'll try to understand everything
I will make the right decision.
I. And now we have not just a lesson, but a cosmic lesson. We'll take a trip to the stars. In flight, we will repeat division, remember how numbers are called when multiplying, adding, subtracting.
And in order for the flight to be successful, you need to listen carefully, think, and count correctly.
But first you need to get permission to take off.
So: we give only the answer.
Difference between 60 and 8 (52)
1st term 32, 2nd term 8 – sum (40)
96 decrease by 90 (6)
The sum of the numbers 16 and 12 (28)
37 increase by 1 (38)
The number 27 contains 3 des and 7 units? (2 d 7 e)
Is the number 38 in the number line between the numbers 37 and 40? (37.39)
7 dec. Is it 70? Yes
5 dec. Is it 15? No.
What are the names of the numbers with + / with -
We did a good job, but tell me, what actions did you repeat?
(+ and -)
Take your seats, check your flight readiness.Reading.
We fly to other planets
We announce it to you.
II. While our rocket is picking up speed, open the logbooks and write down the date of the flight.
We are already in place and flew to the 1 star "Hurry". Here we are waiting for tempting tasks:
5 ,10,11,15,20
40,30, 19 ,20,80 what number is extra?
22,23, 42 ,25,26
40,42,44,46…
35,40,45,50... what number comes next?
10,20,30,40…
And this work must be done quickly and accurately. The task is to solve and test.
38+27 52-29 63-44 51+29 91-55
What action will check +, -.
The guys tried very hard, I liked it, we will not linger here, we will continue the flight.Fizminutka
Together stood up once, 2.3
We are now heroes
We put our hands over our eyes.
Let's set our legs strong.
turned to the right,
Let's look majestically
And to the left, too.
Look from under the palms.
And to the right and more
Over the right shoulder.
III. We did not notice how we flew up to the star "Divide-ka". Let's remember what the numbers are called when dividing?
What is the name of the number we are dividing? (dividend)
What is the name of the number we divide by? (divider)
What is the result of division called? (private)
Let's write in logbook: Divisible 10, divisor 2. What do you need to find? (private)
And we will look for the private with the help of a drawing.
Who will draw?
O O O O O O O O O O
(rocket →)
how many circles we group (2 each)
sk. times 2 is contained in 10? (Five times)
So what is the quotient? (5)
And now look around, look left, right, up. What about our rocket?
In my opinion, she lost control and urgently needs to make calculations. Who will help us? (lay out the cards on the board)
12:3 8:2 6:3
And here is the expression itself:
12:2
Let's repeat.
What is a number called when divided.
What is the result of division?
(Seated exercise)
Stretched, raised the right, left shoulder.
IV. We flew to the star "Reserve". We need to check our stocks:
On the flight we took 5 bottles of lemonade, 1 liter each. How many liters of lemonade did you take? (10l)
And they also took 3 boxes of cookies, 2 kg each. How many kilograms of cookies did you take.
We thought to take 20 large chumps and 10 small ones on the flight. When they found out what it was, they all threw it away. How many chryamziks were thrown out?
Let's split into crews. What color is the star on your desk?
orange
Take your seats. Before work, guess what it is?
Grandfather sits wearing 100 fur coats
Who undresses him, sheds tears
Yes, it's an onion. Do you know what an onion is? Ancient Russia was considered the best remedy from diseases? And in Ancient Greece- a sacred plant. And in Germany, heroes were decorated with onion flowers. Let's take it on a flight?
And what's that?The child grew up did not know diapers,
Became an old man
100 diapers on it.
Of course it's cabbage. It has long been used as a remedy for insomnia and headaches. Her juice was smeared on the wounds.
Shall we take cabbage with us?
And now let's get down to business.
Time has gone.
15 bulbs were planted 3 in a row. How many rows did you get?
15 heads of cabbage were planted equally in 3 rows. How many heads of cabbage in each row.
2) Give a solution.
What did you notice?
(the solution is one, but we are looking for a different one)
We decided, we decided
Something we are very tired,
We are sinking now
Let's clap our hands
Once - swear
Let's get up quickly
Let's smile
Let's sit quietly.
And what's that? Unidentified objects are approaching us, in order to avoid a collision, we urgently need to find out their parameters.
∆ O
What objects did you see?
On what basis do we group?
by color
to size
in form
Name them.
What else is this? Some weird faces.
From what geom. figures consist?
What is extra?
Very well.
We have avoided a collision, and our journey is coming to an end. And in order to safely return back, you need to solve a crossword puzzle.
1) What happens when adding? (sum)
2) What is the name of the number, the result of division? (private)
3) Is it straight and sharp? (corner)
4) The number that is being divided? (dividend)
5) The number by which they divide? (divider)
Very bad score? (unit)
7) What action will we check “+” (subtraction)
Our flight has come to an end. We follow the directions. What word came out. Well done.
Yes, of course we are great.
What did you repeat today?
What did you like?
What seemed difficult?
How do we rate ourselves?
And in order to continue working on division successfully in the next lesson and write well independent work you need to fix the material at home.
Let's write the task:
p.54 tab. Number 3.
The lesson is over.
The division of natural numbers, especially multi-valued ones, is conveniently carried out by a special method, which is called division by a column (in a column). You can also see the name corner division. Immediately, we note that the column can be carried out both division of natural numbers without a remainder, and division of natural numbers with a remainder.
In this article, we will understand how division by a column is performed. Here we will talk about the writing rules, and about all intermediate calculations. First, let us dwell on the division of a multi-valued natural number by a single-digit number by a column. After that, we will focus on cases where both the dividend and the divisor are multi-valued natural numbers. The whole theory of this article is provided with characteristic examples of division by a column of natural numbers with detailed explanations of the solution and illustrations.
Page navigation.
Rules for recording when dividing by a column
Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results when dividing natural numbers by a column. Let's say right away that it is most convenient to divide in a column in writing on paper with a checkered line - so there is less chance of going astray from the desired row and column.
First, the dividend and the divisor are written in one line from left to right, after which a symbol of the form is displayed between the written numbers. For example, if the dividend is the number 6 105, and the divisor is 5 5, then their correct notation when divided into a column will be:
Look at the following diagram, which illustrates the places for writing the dividend, divisor, quotient, remainder, and intermediate calculations when dividing by a column.
It can be seen from the above diagram that the desired quotient (or incomplete quotient when dividing with a remainder) will be written below the divisor under the horizontal line. And intermediate calculations will be carried out below the dividend, and you need to take care of the availability of space on the page in advance. In doing so, the following rule should be followed: more difference in the number of characters in the entries of the dividend and divisor, the more space is required. For example, when dividing a natural number 614,808 by 51,234 by a column (614,808 is a six-digit number, 51,234 is a five-digit number, the difference in the number of characters in the records is 6−5=1), intermediate calculations will require less space than when dividing numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3 ). To confirm our words, we present the completed records of division by a column of these natural numbers:
Now you can go directly to the process of dividing natural numbers by a column.
Division by a column of a natural number by a single-digit natural number, algorithm for dividing by a column
It is clear that dividing one single-digit natural number by another is quite simple, and there is no reason to divide these numbers into a column. However, it will be useful to practice the initial skills of division by a column on these simple examples.
Example.
Let us need to divide by a column 8 by 2.
Solution.
Of course, we can perform division using the multiplication table, and immediately write down the answer 8:2=4.
But we are interested in how to divide these numbers by a column.
First, we write the dividend 8 and the divisor 2 as required by the method:
Now we start to figure out how many times the divisor is in the dividend. To do this, we successively multiply the divisor by the numbers 0, 1, 2, 3, ... until the result is a number equal to the dividend (or a number greater than the dividend, if there is a division with a remainder). If we get a number equal to the dividend, then we immediately write it under the dividend, and in place of the private we write the number by which we multiplied the divisor. If we get a number greater than the divisible, then under the divisor we write the number calculated at the penultimate step, and in place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.
Let's go: 2 0=0 ; 2 1=2; 2 2=4 ; 2 3=6 ; 2 4=8 . We got a number equal to the dividend, so we write it under the dividend, and in place of the private we write the number 4. The record will then look like this:
The final stage of dividing single-digit natural numbers by a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract numbers above this line in the same way as it is done when subtracting natural numbers with a column. The number obtained after subtraction will be the remainder of the division. If it is equal to zero, then the original numbers are divided without a remainder.
In our example, we get
Now we have a finished record of division by a column of the number 8 by 2. We see that the quotient 8:2 is 4 (and the remainder is 0 ).
Answer:
8:2=4 .
Now consider how the division by a column of single-digit natural numbers with a remainder is carried out.
Example.
Divide by a column 7 by 3.
Solution.
At the initial stage, the entry looks like this:
We begin to find out how many times the dividend contains a divisor. We will multiply 3 by 0, 1, 2, 3, etc. until we get a number equal to or greater than the dividend 7. We get 3 0=0<7
; 3·1=3<7
; 3·2=6<7
; 3·3=9>7 (if necessary, refer to the article comparison of natural numbers). Under the dividend we write the number 6 (it was obtained at the penultimate step), and in place of the incomplete quotient we write the number 2 (it was multiplied at the penultimate step).
It remains to carry out the subtraction, and the division by a column of single-digit natural numbers 7 and 3 will be completed.
So the partial quotient is 2 , and the remainder is 1 .
Answer:
7:3=2 (rest. 1) .
Now we can move on to dividing multi-valued natural numbers by single-digit natural numbers by a column.
Now we will analyze column division algorithm. At each stage, we will present the results obtained by dividing the many-valued natural number 140 288 by the single-valued natural number 4 . This example was not chosen by chance, since when solving it, we will encounter all possible nuances, we will be able to analyze them in detail.
First, we look at the first digit from the left in the dividend entry. If the number defined by this figure is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add the next digit to the left in the dividend record, and work further with the number determined by the two digits in question. For convenience, we select in our record the number with which we will work.
The first digit from the left in the dividend 140,288 is the number 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the dividend record. At the same time, we see the number 14, with which we have to work further. We select this number in the notation of the dividend.
The following points from the second to the fourth are repeated cyclically until the division of natural numbers by a column is completed.
Now we need to determine how many times the divisor is contained in the number we are working with (for convenience, let's denote this number as x ). To do this, we successively multiply the divisor by 0, 1, 2, 3, ... until we get the number x or a number greater than x. When a number x is obtained, then we write it under the selected number according to the notation rules used when subtracting by a column of natural numbers. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (during subsequent passes of 2-4 points of the algorithm, this number is written to the right of the numbers already there). When a number is obtained that is greater than the number x, then under the selected number we write the number obtained at the penultimate step, and in place of the quotient (or to the right of the numbers already there) we write the number by which the multiplication was carried out at the penultimate step. (We carried out similar actions in the two examples discussed above).
We multiply the divisor of 4 by the numbers 0 , 1 , 2 , ... until we get a number that is equal to 14 or greater than 14 . We have 4 0=0<14
, 4·1=4<14
, 4·2=8<14
, 4·3=12<14
, 4·4=16>fourteen . Since at the last step we got the number 16, which is greater than 14, then under the selected number we write the number 12, which turned out at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate paragraph the multiplication was carried out precisely on it.
At this stage, from the selected number, subtract the number below it in a column. Below the horizontal line is the result of the subtraction. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at this point is the very last action that completely completes the division by a column). Here, for your control, it will not be superfluous to compare the result of subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake has been made somewhere.
We need to subtract the number 12 from the number 14 in a column (for the correct notation, you must not forget to put a minus sign to the left of the subtracted numbers). After the completion of this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with a divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next item.
Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we did not write zero), we write down the number located in the same column in the record of the dividend. If there are no numbers in the record of the dividend in this column, then the division by a column ends here. After that, we select the number formed under the horizontal line, take it as a working number, and repeat with it from 2 to 4 points of the algorithm.
Under the horizontal line to the right of the number 2 already there, we write the number 0, since it is the number 0 that is in the record of the dividend 140 288 in this column. Thus, the number 20 is formed under the horizontal line.
We select this number 20, take it as a working number, and repeat the actions of the second, third and fourth points of the algorithm with it.
We multiply the divisor of 4 by 0 , 1 , 2 , ... until we get the number 20 or a number that is greater than 20 . We have 4 0=0<20
, 4·1=4<20
, 4·2=8<20
, 4·3=12<20
, 4·4=16<20
, 4·5=20
. Так как мы получили число, равное числу 20
, то записываем его под отмеченным числом, а на месте частного, справа от уже имеющегося там числа 3
записываем число 5
(на него производилось умножение).
We carry out subtraction by a column. Since we subtract equal natural numbers, then, due to the property of subtracting equal natural numbers, we get zero as a result. We do not write down zero (since this is not yet the final stage of dividing by a column), but we remember the place where we could write it down (for convenience, we will mark this place with a black rectangle).
Under the horizontal line to the right of the memorized place, we write down the number 2, since it is she who is in the record of the dividend 140 288 in this column. Thus, under the horizontal line we have the number 2 .
We take the number 2 as a working number, mark it, and once again we will have to perform the steps from 2-4 points of the algorithm.
We multiply the divisor by 0 , 1 , 2 and so on, and compare the resulting numbers with the marked number 2 . We have 4 0=0<2
, 4·1=4>2. Therefore, under the marked number, we write the number 0 (it was obtained at the penultimate step), and in place of the quotient to the right of the number already there, we write the number 0 (we multiplied by 0 at the penultimate step).
We perform subtraction by a column, we get the number 2 under the horizontal line. We check ourselves by comparing the resulting number with the divisor 4 . Since 2<4
, то можно спокойно двигаться дальше.
Under the horizontal line to the right of the number 2, we add the number 8 (since it is in this column in the record of the dividend 140 288). Thus, under the horizontal line is the number 28.
We accept this number as a worker, mark it, and repeat steps 2-4 of paragraphs.
There shouldn't be any problems here if you've been careful up to now. Having done all the necessary actions, the following result is obtained.
It remains for the last time to carry out the actions from points 2, 3, 4 (we provide it to you), after which you will get a complete picture of dividing natural numbers 140 288 and 4 into a column:
Please note that the number 0 is written at the very bottom of the line. If this were not the last step of dividing by a column (that is, if there were numbers in the columns on the right in the record of the dividend), then we would not write this zero.
Thus, looking at the completed record of dividing the multi-valued natural number 140 288 by the single-valued natural number 4, we see that the number 35 072 is private (and the remainder of the division is zero, it is on the very bottom line).
Of course, when dividing natural numbers by a column, you will not describe all your actions in such detail. Your solutions will look something like the following examples.
Example.
Perform long division if the dividend is 7136 and the divisor is a single natural number 9.
Solution.
At the first step of the algorithm for dividing natural numbers by a column, we get a record of the form
After performing the actions from the second, third and fourth points of the algorithm, the record of division by a column will take the form
Repeating the cycle, we will have
One more pass will give us a complete picture of division by a column of natural numbers 7 136 and 9
Thus, the partial quotient is 792 , and the remainder of the division is 8 .
Answer:
7 136:9=792 (rest 8) .
And this example demonstrates how long division should look like.
Example.
Divide the natural number 7 042 035 by the single digit natural number 7 .
Solution.
It is most convenient to perform division by a column.
Answer:
7 042 035:7=1 006 005 .
Division by a column of multivalued natural numbers
We hasten to please you: if you have well mastered the algorithm for dividing by a column from the previous paragraph of this article, then you already almost know how to perform division by a column of multivalued natural numbers. This is true, since steps 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first step.
At the first stage of dividing into a column of multi-valued natural numbers, you need to look not at the first digit on the left in the dividend entry, but at as many of them as there are digits in the divisor entry. If the number defined by these numbers is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the record of the dividend. After that, the actions indicated in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.
It remains only to see the application of the algorithm for dividing by a column of multi-valued natural numbers in practice when solving examples.
Example.
Let's perform division by a column of multivalued natural numbers 5562 and 206.
Solution.
Since 3 characters are involved in the record of the divisor 206, we look at the first 3 digits on the left in the record of the dividend 5 562. These numbers correspond to the number 556. Since 556 is greater than the divisor 206, we take the number 556 as a working one, select it, and proceed to the next stage of the algorithm.
Now we multiply the divisor 206 by the numbers 0 , 1 , 2 , 3 , ... until we get a number that is either equal to 556 or greater than 556 . We have (if the multiplication is difficult, then it is better to perform the multiplication of natural numbers in a column): 206 0=0<556
, 206·1=206<556
, 206·2=412<556
, 206·3=618>556 . Since we got a number that is greater than the number 556, then under the selected number we write the number 412 (it was obtained at the penultimate step), and in place of the quotient we write the number 2 (since it was multiplied at the penultimate step). The column division entry takes the following form:
Perform column subtraction. We get the difference 144, this number is less than the divisor, so you can safely continue to perform the required actions.
Under the horizontal line to the right of the number available there, we write the number 2, since it is in the record of the dividend 5 562 in this column:
Now we work with the number 1442, select it, and go through steps two through four again.
We multiply the divisor 206 by 0 , 1 , 2 , 3 , ... until we get the number 1442 or a number that is greater than 1442 . Let's go: 206 0=0<1 442
, 206·1=206<1 442
, 206·2=412<1 332
, 206·3=618<1 442
, 206·4=824<1 442
, 206·5=1 030<1 442
, 206·6=1 236<1 442
, 206·7=1 442
. Таким образом, под отмеченным числом записываем 1 442
, а на месте частного правее уже имеющегося там числа записываем 7
:
We subtract by a column, we get zero, but we don’t write it down right away, but only remember its position, because we don’t know if the division ends here, or we will have to repeat the steps of the algorithm again:
Now we see that under the horizontal line to the right of the memorized position, we cannot write down any number, since there are no numbers in the record of the dividend in this column. Therefore, this division by a column is over, and we complete the entry:
- Maths. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
- Maths. Any textbooks for 5 classes of educational institutions.
Sections: Maths
Class: 5
Topic: Division with remainder.
Lesson Objectives:
Repeat division with a remainder, derive a rule on how to find the dividend when dividing with a remainder, and write it as a literal expression;
- develop attention, logical thinking, mathematical speech;
- fostering a culture of speech, perseverance.
During the classes
The lesson is accompanied by a computer presentation. (Application)
I. Organizing time
II. Verbal counting. Lesson topic message
After solving the examples and filling in the table, you will be able to read the topic of the lesson.
On the desk:
Read the topic of the lesson.
They opened notebooks, wrote down the date, the topic of the lesson. (Slide 1)
III. Work on the topic of the lesson
Decide verbally. (Slide 2)
1. Read the expressions:
30: 5
103: 10
34: 5
60: 7
47: 6
131: 11
42: 6
What two groups can they be divided into? Write down and solve those in which the division is with a remainder.
2. Let's check. (Slide 3)
No remainder: |
With the remainder: |
|
30: 5 |
103: 10 = 10 (rest 3) |
Can you tell me how you did division with a remainder?
Not always one natural number is divisible by another number. But you can always perform division with a remainder.
What does it mean to divide with the remainder? To answer this question, let's solve the problem. ( slide 4)
4 grandchildren came to visit their grandmother. Grandmother decided to treat her grandchildren with sweets. There were 23 candies in the vase. How many sweets will each grandchild get if the grandmother offers to share the candies equally?
Let's reason.
How many candies does grandma have? (23)
How many grandchildren came to visit their grandmother? (4)
What needs to be done according to the condition of the task? (Candies must be divided equally, 23 must be divided by 4; 23 is divided by 4 with a remainder; in the quotient it will be 5, and the remainder will be 3.)
How many sweets will each grandchild get? (Each grandchild will get 5 candies, and 3 candies will remain in the vase.)
Let's write down the solution. (Slide 5)
23: 4=5 (rest 3)
What is the name of the number that is being divided? (Divisible.)
What is a divider? (Number by which to divide.)
What is the result of division with a remainder called? (Incomplete quotient.)
Name the dividend, divisor, partial quotient and remainder in our solution (23 is the dividend, 4 is the divisor, 5 is the partial quotient, 3 is the remainder.)
Guys, think and write down how to find the dividend 23, knowing the divisor, incomplete quotient and remainder?
Let's check.
Guys, let's formulate a rule on how to find the dividend if the divisor, incomplete quotient and remainder are known.
Rule. (Slide 6)
The dividend is equal to the product of the divisor and the incomplete quotient, added with the remainder.
a = sun + d , a - dividend, c - divisor, c - partial quotient, d - remainder.
When division with a remainder is performed, what should we remember?
That's right, the remainder is always less than the divisor.
And if the remainder is zero, the dividend is divisible by the divisor without a remainder, completely.
IV. Consolidation of the studied material
Slide 7
Find the dividend if:
A) the partial quotient is 7, the remainder is 3, and the divisor is 6.
B) the incomplete quotient is 11, the remainder is 1, and the divisor is 9.
C) the partial quotient is 20, the remainder is 13, and the divisor is 15.
V. Working with the textbook
1.
Working on a task.
2.
Formulating a solution to a problem.
№ 516 (The student solves the problem at the blackboard.)
20 x 10: 18 = 11 (rest 2)
Answer: 11 parts of 18 kg each can be cast from 10 ingots, 2 kg of cast iron will remain.
№ 519 (Workbook, p. 52 No. 1.)
slide 8, 9
The first task is done by the student at the blackboard. The second and third - students perform independently with self-examination.
We solve problems verbally. (Slide 10)
VI. Lesson summary
There are 17 students in your class. You were lined up. It turned out several lines of 5 students and one incomplete line. How many full lines did it turn out and how many people are in an incomplete line?
Your class in the physical education lesson was again lined up. This time it turned out 4 identical full lines and one incomplete? How many people are in each line? And in incomplete?
We answer questions:
Can the remainder be greater than the divisor? Can the remainder be equal to the divisor?
How to find the dividend by the incomplete quotient, divisor and remainder?
What are the remainders when divided by 5? Give examples.
How to check if division with remainder is correct?
Oksana thought of a number. If this number is increased by 7 times and 17 is added to the product, then it will be 108. What number did Oksana think of?
VII. Homework
Item 13, No. 537, 538, workbook, p. 42, no. 4.
Bibliography
1. Mathematics: Proc. for 5 cells. general education institutions / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - 9th ed., stereotype. – M.: Mnemozina, 2001. – 384 p.: ill.
2. Mathematics. Grade 5 Workbook number 1. natural numbers / V.N. Rudnitskaya. – 7th ed. – M.: Mnemozina, 2008. – 87 p.: ill.
3. Chesnokov A.S., Neshkov K.I. Didactic materials in mathematics for grade 5. - M. : Classics Style, 2007. - 144 p.: ill.
Sections: Maths
Class: 6
Lesson Objectives:
1. Educational: repetition, generalization and testing of knowledge on the topic: “Divisibility of natural numbers”; development of basic skills.
2. Developing: to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.
3. Educational: through a lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, independence.
Lesson objectives:
To form the ability to apply the concept of divisors and multiples; develop thinking and elements of creative activity; apply the signs of divisibility in the simplest situations; finding GCD and LCM numbers, develop observation and logical thinking.
Lesson type- combined.
Lesson Form- A lesson with computer support.
Equipment:
1. Board and chalk.
2. Computer and projector.
3. Paper version of all tasks.
During the classes.
Numbers rule the world.
Pythagoras.
1. Organizational moment.
2. Communication of the purpose of the lesson.
3. Actualization of basic knowledge.
1. What is called the divisor of a number a?
2. What is called a multiple of a number a?
3. Is there a greatest multiple?
4. Formulate signs of divisibility?
5. Which numbers are called prime and which are composite?
(Student report about Pythagoras, about Eratosthenes, about Euclid)
Historical information:
Euclid - an ancient Greek scientist (365 - 300 BC). Very little is known about the life of this great scientist. He lived and worked in Alexandria, a city founded by Alexander the Great. There are many legends associated with the name of Euclid. One of them tells that King Ptolemy asked Euclid: “Is there no shorter way to the knowledge of geometry?”, To which the scientist replied: “There is no royal road to geometry!”. Euclid did a lot of number theory: it was he who proved that there are infinitely many prime numbers. The algorithm for finding GCD of two numbers is called Euclid's algorithm. The ancient Greek mathematician Euclid, in his book "Principles", which was for two thousand years the main textbook of mathematics, proved that there are infinitely many prime numbers, i.e. behind every prime number there is another prime number. |
Pythagoras (6th century BC) and his students studied the divisibility of numbers. The number equal to the sum of all its divisors (without the number itself), they called the perfect number. For example, the number 6 (6 = 1 + 2 + 3) , 28 (28 = 1 + 2 + 4 + 7 + 14) are perfect. The next perfect numbers are 496, 8128, 33550336 The Pythagoreans only knew the first three perfect numbers. The fourth 8128 became known in the 1st century BC. The fifth number 33550336 was found in the 15th century. By 1983, 27 perfect numbers were already known. But until now, scientists do not know whether there is an odd perfect number, whether there is a largest perfect number. The interest of ancient mathematicians in prime numbers is due to the fact that any natural number greater than 1 is either a prime number, or can be composed as a product of prime numbers: 14 = 2∙ 7, 16 = 2∙2 ∙2∙2 The question arises: does the last (largest) prime number exist? |
Problem: Think of a prime number. The next natural number is also prime. What numbers are we talking about?
Answer: 2.3.
6. What numbers are called relatively prime?
7. Explain how to find GCD (LCC) of two numbers.
(Student's message about finding the GCD of two numbers)
One day, the numbers 24 and 60 argued about how to find the GCD. The number 24 stated that you first need to find common numbers among all divisors, and then choose the largest number from them. And the number 60 objected:
- Well, what are you! I don't like this way. I have too many divisors, and when listing them, I can skip some. What if it turns out to be the biggest? No, I don't like this way. And they decided to seek help from the master of DELENCHESKII sciences. And the master answered them:
- Yes, 24, your way of finding GCD numbers can be used, but it's not always convenient. And you can find the NOD in another way.
It is necessary to decompose 24 and 60 into prime factors.
24 | 2 |
12 | 2 |
6 | 2 |
3 | 3 |
1 |
60 | 2 |
30 | 2 |
15 | 3 |
5 | 5 |
1 |
24 = 2³ ∙ 3
60 = 2² ∙ 3 ∙ 5
You need to take the common divisors of numbers with a smaller exponent.
GCD (24; 60) \u003d 2² ∙ 3 \u003d 12.
And to find the LCM of two numbers you need:
- Decompose into prime factors;
- Write out all the prime factors that are included in the first number and in the second number with the largest exponent.
Means:
24 = 2³ ∙ 3 60 = 2² ∙ 3 ∙ 5 NOC (24; 60) = 2³∙ 3 ∙ 5 = 120.
Compiled by teacher of the Department of Higher Mathematics Ishchanov T.R.
Lesson number 1. Elements of combinatorics
Theory.
Multiplication rule: if from some finite set the first object (element) can be chosen in ways, and the second object (element) in ways, then both objects ( and ) in the specified order can be chosen in ways.
Addition rule: if some object can be chosen in ways, and the object can be chosen in ways, and the first and second ways do not intersect, then any of the objects ( or ) can be chosen in ways.
practical material.
1. (6.1.44. L) How many different three-digit numbers can be made from the numbers 0, 1, 2, 3, 4 if:
a) numbers cannot be repeated;
b) the numbers can be repeated;
c) numbers must be even (numbers can be repeated);
d) the number must be divisible by 5 (numbers cannot be repeated)
(Answer: a) 48 b) 100 c) 60 d) 12)
2. (6.1.2.) How many numbers containing at least three different digits can be made from the digits 3, 4, 5, 6, 7? (Answer: 300.)
3. (6.1.39) How many four-digit numbers can be formed so that any two adjacent digits are different? (Answer: 6561)
Theory. Let a set consisting of n distinct elements be given.
An arrangement of n elements by k elements (0?k?n) is any ordered subset of a given set that contains k elements. Two arrangements are different if they differ from each other either in the composition of the elements or in the order in which they appear.
The number of placements of n elements by k is denoted by a symbol and is calculated by the formula:
where n!=1·2·3·…·n , and 1!=1,0!=1.
practical material.
4. (6.1.9 L.) Compose various arrangements of two elements from the elements of the set A=(3,4,5) and count their number. (Answer: 6)
5. (6.1.3 K) In how many ways can three prizes be distributed among 16 competitors? (Answer: 3360)
6. (6.1.11. K) How many five-digit numbers are there, all the digits of which are different? Hint: take into account the fact that numbers like 02345, 09782, etc. do not count as 5 digits. (Answer: 27216)
7. (6.1.12.L.) In how many ways can a tricolor striped flag (three horizontal stripes) be made if there is matter of 5 different colors? (Answer: 60.)
Theory. A combination of n elements by k elements (0?k?n) is any subset of the given set that contains k elements.
Any two combinations differ from each other only in the composition of the elements. The number of combinations of n elements by k is denoted by a symbol and is calculated by the formula:
practical material.
8.(6.1.20.) Make various combinations of two elements from the elements of the set A=(3,4,5) and count their number. (Answer: 3.)
9. (6.1.25.) A group of tourists from 12 boys and 7 girls chooses 5 people by lot to cook dinner. How many ways are there in which this "five" will get:
a) only girls b) 3 boys and 2 girls;
c) 1 boy and 4 girls; d) 5 boys; e) tourists of the same sex.
(Answer: a) 21; b) 4620; c) 420; d) 792; e) 813.)
Theory. A permutation of n elements is an arrangement of n elements by n elements. Thus, to indicate one or another permutation of a given set of n elements means to choose a certain order of these elements. Therefore, any two permutations differ from each other only in the order of the elements.
The number of permutations of n elements is denoted by a symbol and is calculated by the formula:
practical material.
10.(6.1.14.L) Compose different permutations from the elements of the set A=(5;8;9). (Answer: 6)
11.(6.1.15.L) In how many ways can a ten-volume set of works by D. London be arranged on a bookshelf, arranging them:
a) randomly
b) so that 1, 5, 9 volumes stand side by side (in any order);
c) so that 1, 2, 3 volumes stand side by side (in any order).
(Answer: a) 10! b) 8!?3! in) )
12. (1.6.16.L.) There are 7 chairs in the room. In how many ways can 7 guests be placed on them? 3 guests? (Answer: 5040; 210)
Selection scheme with return.
Theory. If, in an ordered selection of k elements out of n, elements are returned back, then the resulting samples are arrangements with repetitions. The number of all placements with repetitions of n elements by k is denoted by the symbol and is calculated by the formula:
If, when selecting k elements from n, the elements are returned back without subsequent ordering (thus, the same elements can be taken out several times, i.e., repeated), then the resulting samples are combinations with repetitions. The number of all combinations with repetitions of n elements by k is denoted by a symbol and is calculated by the formula:
practical material.
13.(6.1.29.) From elements (numbers) 2, 4, 5, make up all placements and combinations with repetitions of two elements. (Answer: 9; 6)
14. (6.1.31.L.) Five people entered the elevator on the 1st floor of a nine-story building. In how many ways can passengers exit the elevator at the desired floors? (Answer: )
15. (6.1.59.L.) There are 7 types of cakes in the confectionery. In how many ways can you buy in it: a) 3 cakes of the same type; b) 5 cakes? (Answer: a) 7; b) 462)
Theory. Let a set of n elements have k different types of elements, while the 1st type of elements is repeated once, the 2nd - times, . . . , kth - times, and . Then the permutations of the elements of this set are permutations with repetitions.
The number of permutations with repetitions (sometimes it refers to the number of partitions of a set) of n elements is denoted by a symbol and is calculated by the formula:
practical material.
16.(6.1.32.) How many different "words" (a "word" means any combination of letters) can be formed by rearranging the letters in the word AHA? MISSISSIPPI?
Solution.
In general, three letters can be used to make various three-letter "words". In the word AGA, the letter A is repeated, and the rearrangement of the same letters does not change the "word". Therefore, the number of permutations with repetitions is less than the number of permutations without repetitions as many times as it is possible to permute repeating letters. In this word, two letters (1st and 3rd) are repeated; therefore, there are as many different permutations of three-letter "words" from the letters of the word AGA: . However, the answer can be obtained more simply:. Using the same formula, we will find the number of eleven-letter "words" when permuting the letters in the word MISSISSIPPI. Here (4 letters S), (4 letters I), so
17.(6.1.38.L.) How many different permutations of letters are there in the word TREATMENT? And in the "word" AAAAAAAAAA? (Answer: 420;210)